3.1.14 Sets

Questions concerning sets are by far the easiest problems on the number sense tests. The only topics that are actively questioned are the definitions of intersection, union, compliment, and subsets. Let sets $A = \{M, E, N, T, A, L\}$ and $B = \{M, A, T, H\}$ then:

• Intersection: The intersection between A and B (notated as $C = A \cap B$) is defined to be elements which are in both sets A and B. So in our case $C = A \cap B = \{M, A, T\}$ which consists of 3 elements.
• Union: The union between A and B (notated as $D = A \cup B$) is defined to be a set which contains all elements in A and all elements in B. So $D = A \cup B = \{M, E, N, T, A, L, H\}$ which consists of 7 elements.
• Complement: Let’s solely look at set A and define a new set $E = \{T, E, N\}$. Then the complement of E (notated a variety of ways, typically $\bar{E}$ of $E'$) with respect to Set A consists of simply all elements in A which aren’t in E. So $\bar{E} = \{M, A, L\}$, which consists of three elements.
• Subsets: The number of possible subsets of a set is $2^n$ where $n$ is the number of elements in the set. The number of proper subsets consists of all subsets which are strictly in the set. The result is that this disregards the subset of the set itself. So the number of proper subsets is $2^n - 1$. So in our example, the number of subsets of A is $2^7 = 128$ and the number of proper subsets is $2^7 - 1 = 127$. Another way to ask how many different subsets a particular set has is asking how many elements are in a set’s Power Set. So the number of elements in the Power Set of B is simply $2^4 = 16$.

The following are questions concerning general set theory on the number sense test:

Problem Set 3.1.14